A rigidity theorem for self-shrinkers of MCF. V. Palmer, UJI, - - PowerPoint PPT Presentation

A rigidity theorem for self-shrinkers of MCF.

• V. Palmer, UJI, Castell´
• joint work with:
• V. Gimeno, UJI

30.10.2019 Symmetry and shape Celebrating the 60th birthday of Prof. J. Berndt Santiago de Compostela, (Spain)

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

• Index. I

1 Part I. Introduction:

Introduction: Definition of soliton of MCF Introduction: A classification (gap) theorem of proper self-shrinkers

• f MCF.

Introduction: When the sphere separates a soliton.

2 Part II. Our results: a refinement of this classification,

(Theorems 1 and 2)

3 Part III. Proof of our results:

Minimal immersions into the sphere and self-shrinkers Proof of Theorem 1 Proof of Theorem 2

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part I. Introduction: Definition of soliton of MCF. 1/25

Definition 1 A complete isometric immersion X : Σn → Rn+m is a λ-soliton of the MCF with respect to 0 ∈ Rn+m, (λ ∈ R), if and only if

• H = −λX ⊥

where X ⊥ stands for the normal component of X and H is the mean curvature vector of the immersion X. Definition 2 A λ-soliton for the MCF with respect to 0 ∈ Rn+m is called a self-shrinker if and only if λ ≥ 0. It is called a self-expander if and

• nly if λ < 0.
• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part I. Introduction: Definition of soliton of MCF. 2/25

Remark 3 Given a complete immersion X : Σn → Rn+m satisfying

• H = −λX ⊥

the family of homothetic immersions Xt = √ 1 − 2λtX satisfies the equation of the MCF

• ( ∂

∂t X(p, t))⊥

=

• H(p, t) ∀p ∈ Σ, ∀ t ∈ [0, T)

X(p, 0) = X0(p), ∀p ∈ Σ so X becomes the 0-slice of the family {Xt}∞

t=0 of solutions of equation above.

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part I. Introduction: Definition of soliton of MCF. 3/25

Example 4 A compact λ-self-shrinker X : Σn → Rn+m is Σ = Sn+m−1 √ n

λ

( 0) Complete non-compact self-shrinkers:

Γ × Rn−1 ⊆ Rn+m, where Γ is an Abresch-Langer curve Sk(

• k

λ) × Rn−k ⊆ Rn+m, generalized cylinders

Σ = Rn ⊆ Rn+m is an Euclidean subespace, (case λ = 0).

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part I. Introduction: A classification (gap) theorem of proper self-shrinkers of MCF. 4/25

• H. D. Cao and H. Li proved the following classification result for properly

immersed self-shrinkers

• Theorem. H. D. Cao and H. Li, Calc. Var. 46 (2013)

Let X : Σn → Rn+m be a complete and proper λ-self-shrinker, with bounded norm of the second fundamental form by ARn+m

Σ

2 ≤ λ, Then Σ is one of the following:

1

Σ is a round sphere Sn( n

λ), (and hence ARn+m Σ

2 = λ).

2

Σ is a cylinder Sk(

• k

λ) × Rn−k, (and hence ARn+m Σ

2 = λ).

3

Σ is an hyperplane, (and hence ARn+m

Σ

2 = 0).

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part I. Introduction: When the sphere separates a soliton.

5/25

Definition 5 Let X : Σn → Rn+m be an isometric immersion. We say that the sphere Sn+m−1 √ n

λ

( 0) separates X(Σ) if and only if X(Σ) ∩ Bn+m √ n

λ

( 0) = ∅ and X(Σ) ∩

• Rn+m \ ¯

Bn+m √ n

λ

( 0)

• = ∅.

Namely, there exists p, q ∈ Σ such that r

0(p) = distRn+m(

0, X(p)) = X(p) < n

λ and

r

0(q) = X(q) > n λ.

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part I. Introduction: When the sphere separates a soliton.

6/25

Definition 6 Let X : Σn → Rn+m be an isometric immersion. We say that the sphere Sn+m−1 √ n

λ

( 0) does not separate X(Σ) if and only if X(Σ) ∩ Bn+m √ n

λ

( 0) = ∅ or X(Σ) ∩

• Rn+m \ ¯

Bn+m √ n

λ

( 0)

• = ∅.

Namely, ∀p ∈ Σ, we have r

0(p) = X(p) ≤ n λ or

r

0(p) = X(q) ≥ n λ

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part I. Introduction: When the sphere separates a soliton.

7/25

A cylinder separated by one sphere A cylinder separated by one sphere A cylinder non sep- arated by one sphere

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part II. Our results.

8/25

Theorem 1. V. Gimeno and V. P., JGA, 2019 Let X : Σn → Rn+m be a complete and proper λ-self-shrinker. Let us suppose that the sphere Sn+m−1 √ n

λ

( 0) does not separate X(Σ). Then Σn is compact and X : Σ → Sn+m−1( n

λ) is a minimal

immersion.

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part II. Our results.

9/25

Corollary 1. M. P. Cavalcante-J.M. Espinar, Bull. London Math.

• Soc. 48 (2016), V. Gimeno and V. P., JGA, 2019

Let X : Σn → Rn+1 be a complete, connected and proper λ-self-shrinker. Let us suppose that the sphere Sn+m−1 √ n

λ

( 0) does not separate X(Σ). Then, Σn is isometric to Sn n

λ

• Sketch of proof

No separation by the sphere implies, (Theorem 1), that X : Σ → Sn+m−1( n

λ) is a minimal immersion.

The local isometry X : Σn → Sn n

λ

• among

connected/simply connected spaces becomes a Riemannian covering and hence, an isometry.

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part II. Our results.

10/25

Theorem 2. V. Gimeno and V. P., JGA, 2019 Let X : Σn → Rn+m, (m ≥ 2), be a complete and proper λ-self-shrinker, such that: i) The sphere Sn+m−1 √ n

λ

( 0) does not separate X(Σ). ii) The second fundamental form of Σ is bounded by ARn+m

Σ

2 < 5 3λ Then, Σn is isometric to Sn n

λ

• .
• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part II. Our results.

11/25

We would like to emphasize the analogy of our notion of“separation by spheres”with the notion of“separation by planes”used in the Halfspace theorem for self-shrinkers. Halfspace theorem for self-shrinkers, see M. P. Cavalcante-J.M. Espinar, Bull. London Math. Soc. 48 (2016) and S. Pigola-M. Rimoldi, Ann. Global Analysis 45 (2014) Let Pn be an hyperplane in Rn+1 passing through the origin. The only properly immersed self-shrinker Σn contained in one of the closed half-space determined by P is Σ = P. In this sense, Corollary 1 above could be stated as: Theorem, (Corollary 1) The only properly immersed and connected self-shrinker Σn contained in one of the closed domains determined by the sphere Sn √ n

λ (

0), is Σn = Sn √ n

λ (

0)

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part III. Proof of our results. Minimal immersions into the sphere and self-shrinkers.

12/25

Proposition 7 (K. Smoczyk, Int. Math. Res. Not. 48 (2005)) Let X : Σn → Sn+m−1(R) be a complete spherical immersion. Then, the following affirmations are equivalent:

1 X : Σn → Sn+m−1(R) is a minimal immersion into

Sn+m−1(R).

2 X is a λ- self-shrinker with λ =

n R2 , i.e., R = n λ

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part III. Proof of our results. Minimal immersions into the sphere and self-shrinkers.

13/25

Proof of Proposition 7 To see 1) ⇒ 2), use the equation

• HΣ⊆Rn+m =

HΣ⊆Sn+m−1(R) − n R2 X = − n R2 X = − n R2 X ⊥ To see 2)⇒ 1), use that X is a λ-self-shrinker and the extrinsic distance function r

0(p) := distRn+m(

0, X(p)) defined on Σ. Given F(p) := r 2(p) = X2 = R2 on Σ, apply Lemma 8 Given F : Σ → R, F ∈ C 2(Σ), for all x ∈ Σ such that r(x) > 0, we have ∆ΣF(r(x)) =

• F ′′(r(x))

r2(x)

− F ′(r(x))

r3(x)

• X T2

+ F ′(r(x))

r(x)

• n + X,

HΣ⊆Rn+m

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part III. Proof of our results. Theorem 1.14/25

We are going to prove Theorem 1 Let X : Σn → Rn+m be a complete properly immersed λ-self-shrinker. Let us suppose that the sphere Sn+m−1 √ n

λ

( 0) does not separate X(Σ). Then Σn is compact and X : Σ → Sn+m−1( n

λ) is a minimal

immersion.

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part III. Proof of our results. Theorem 1.

15/25

Proof of Theorem 1 As Sn+m−1 √ n

λ

( 0) does not separate X(Σ) then X(Σ) ⊆ ¯ Bn+m √ n

λ (

0) or X(Σ) ⊆ Rn+m \ Bn+m √ n

λ (

0). Suppose first that X(Σ) ⊆ ¯ Bn+m √ n

λ (

0). Then n

λ ≥ r(p) ∀p ∈ Σ.

Hence X ⊥2 ≤ X2 ≤ n

λ.

Then, compute, (using Lemma 8 above): △Σr 2(x) = 2(n − λX ⊥2) ≥ 0

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part III. Proof of our results. Theorem 1.

16/25

Proof of Theorem 1 As X is proper and Σ = X −1(¯ Bn+m √ n

λ (

0)), then Σ is compact and hence, by Hopf’s Lemma: r 2(x) = R2 ∀x ∈ Σ, so we have the spherical immersion X : Σ → Sn+m−1(R), for some R ≤ n

λ.

As Σ is a λ-soliton for the MCF, then R = n

λ, and

X : Σ → Sn+m−1( n

λ) is minimal by Proposition 7.

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part III. Proof of our results. Theorem 1.

17/25

Proof of Theorem 1 Suppose now that X(Σ) ⊆ Rn+m \ Bn+m √ n

λ (

0). Then n

λ ≤ r(p) ∀p ∈ Σ.

Assume that X(Σ) ⊆ Sn+m−1(R) for any radius R > 0 and that infΣ r > n

λ. We will reach a contradiction.

First, as infΣ r > n

λ, we have that, for any p ∈ Σ

1 − λ n r 2(p) < 0

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part III. Proof of our results. Theorem 1.

18/25

Proof of Theorem 1 Hence, given the extrinsic ball DR = X −1(Bn+m

R

( 0)) = {p ∈ Σ : X(p) < R} ⊆ Σ and integrating

• DR
• 1 − λ

n r 2

• e

λ 2 (R2−r2)dσ < 0

(1) Now, we need the following Lemma 9 Let X : Σn → Rn+m be a complete properly immersed λ-self-shrinker in Rn+m. Let us suppose that X(Σ) ⊆ Sn+m−1(R) for any radius R > 0. Given the extrinsic ball DR = X −1(Bn+m

R

( 0)), if Vol(DR) > 0, we have, for all R > 0: 0 ≤ 1 −

• DR

HΣ2dσ nλ Vol(DR) =

• DR
• 1 − λ

n r 2

e

λ 2 (R2−r2)dσ

Vol(DR) (2)

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part III. Proof of our results. Theorem 1.

19/25

Proof of Theorem 1 Now, applying inequality (1) and Lemma 9, we have 0 ≤ 1 −

• DR

HΣ2dσ nλ Vol(DR) =

• DR
• 1 − λ

n r 2

e

λ 2 (R2−r2)dσ

Vol(DR) < 0 (3) which is a contradiction. Hence, either X(Σ) ⊆ Sn+m−1(R0) for some radius R0 > 0, or infΣ r = n

λ.

In the first case, we have that X : Σ → Sn+m−1(R0) will be a spherical immersion and, by Proposition 7, as Σ is a λ-self-shrinker, then X is minimal and λ =

n R20 , namely, X : Σ → Sn+m−1( n λ) is a minimal

immersion.

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part III. Proof of our results. Theorem 1.

20/25

Proof of Theorem 1 In the second case, if infΣ r = n

λ, then n λ ≤ r(p) for all p ∈ Σ and

hence 1 − λ

n r 2(p) ≤ 0 ∀p ∈ Σ.

Then by inequality (1) and Lemma 9 we have 0 ≤ 1 −

• DR

HΣ2dσ nλ Vol(DR) =

• DR
• 1 − λ

n r 2

e

λ 2 (R2−r2)dσ

Vol(DR) ≤ 0 (4) Therefore, 1 − λ

n r 2(p) = 0 ∀p ∈ Σ, so X(Σ) ⊆ Sn+m−1( n λ), and hence

X : Σ → Sn+m−1( n

λ) is a complete spherical immersion and a

λ-self-shrinker. Then by Proposition 7, Σ is minimal in the sphere Sn+m−1( n

λ).

Finally, as X : Σn → Rn+m is proper, then Σ = X −1(Sn+m−1( n

λ)) is

compact.

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part III. Proof of our results. Theorem 2.

21/25

We are going to prove Theorem 2 Let X : Σn → Rn+m, (m ≥ 2), be a complete and proper λ-self-shrinker, such that: i) The sphere Sn+m−1 √ n

λ

( 0) does not separate X(Σ) ii) The second fundamental form of Σ is bounded by ARn+m

Σ

2 < 5 3λ Then, Σn is isometric to Sn n

λ

• and ARn+m

Σ

2 = λ.

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part III. Proof of our results. Theorem 2.

22/25

Proof of Theorem 2 If the sphere Sn+m−1 √ n

λ

( 0) does not separate X(Σ), then, applying Theorem 1, X : (Σ, g) → (Sn+m−1( n

λ), gSn+m−1(√ n

λ )) is a compact and minimal

immersion, Hence, scaling the metric, X : (Σ, λ

n g) → (Sn+m−1(1), gSn+m−1(1)) realizes

as a minimal immersion, with second fundamental form in the sphere satisfying

• ASn+m−1(1)

Σ

• 2

= n λARn+m

Σ

2 − n (5) Hence, as by hypothesis ARn+m

Σ

2 < 5

3λ, then

• ASn+m−1(1)

Σ

• 2

< 2n 3 (6)

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part III. Proof of our results. Theorem 2.

23/25

Proof of Theorem 2

Case I: Assume that n ≥ 1 and m = 2. Apply following Theorem,

• J. Simons-M.P. Do Carmo-S.S. Chern-S. Kobayashi Rigidity

Theorem

Let ϕ : (Σn, g) → (Sn+1(1), gSn+1(1)) be a compact and minimal isometric immersion. Let us suppose that

• ASn+1(1)

Σ

2 ≤ n. Then 1 either ASn+1(1)

Σ

2 = 0 and (Σn, g) is isometric to Sn(1) 2

• r

ASn+1(1)

Σ

2 = n. Then (Σn, g) is isometric to a generalized Clifford torus Σn = Sk (

• k

n ) × Sn−k (

• n−k

n

) immersed as an hypersurface in Sn+1(1).

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Part III. Proof of our results. Theorem 2.

24/25

Proof of Theorem 2 Case II: Assume that n ≥ 1 and m ≥ 3. Apply following Theorem

• A. M Li and J. Li, Archiv. Math. 58, (1992). Refinement of Simons’ et al.

Theorem Let ϕ : (Σn, g) → (Sn+m−1(1), gSn+m−1(1)) be a compact and minimal isometric immersion, and m ≥ 3. Let us suppose that ASn+m−1(1)

Σ

2 ≤ 2n

3 . Then, 1

either ASn+m−1(1)

Σ

2 = 0 and (Σn, g) is isometric to Sn(1)

2

• r, (in case n = 2 and m = 3),

ASn+m−1(1)

Σ

2 = 4

3 and (Σ2,

g) is isometric to the Veronese surface Σ2 = RP2( √ 3) in S4(1).

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

• END. 25/25

Thank you

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Proof of our results. Lemma 9.

Proof of Lemma 9 Consider r 2 : Σ → R, defined as r 2(p) = X(p)2, where r = distRn+m( 0, ). We have that X = r∇Rn+mr and that X T = r∇Σr Then, applying Lemma 8 to the radial function F(r) = r 2, ∆Σr 2 = 2n + 2r∇Rn+mr, HΣ (7) As r∇Rn+mr, HΣ = −λX ⊥ = −

HΣ2 λ

Then ∆Σr 2 = 2n − 2 HΣ2 λ (8)

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Proof of our results. Lemma 9.

Proof of Lemma 9 Integrating on DR = X −1(Bn+m

R

( 0)) equality above, we have nλ Vol(DR) −

• DR
• HΣ2dσ = λ

2

• DR

∆Σr 2dσ (9) Apply Divergence theorem (unitary normal to ∂DR in Σ, pointed outward is µ =

∇Σr ∇Σr and X T = r∇Σr),

• DR

∆Σr 2dσ =

• ∂DR

∇Σr 2, ∇Σr ∇Σrdµ =

• ∂DR

2r∇Σrdµ = 2

• ∂DR

X Tdµ (10)

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Proof of our results. Lemma 9.

Proof of Lemma 9 Then equation (9) becomes nλ Vol(DR) −

• DR
• HΣ2dσ = λ
• ∂DR

X Tdµ = λ

• ∂DR

r∇Σrdµ = λR

• ∂DR

∇Σrdµ (11) Hence 1 −

• DR

HΣ2dσ nλ Vol(DR) = R n Vol(DR)

• ∂DR

∇Σrdµ ≥ 0 (12)

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• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

Proof of our results. Lemma 9.

Proof of Lemma 9 Applying the divergence theorem on DR to the vector field e− λ

2 r2∇Σr 2,

we obtain

• DR

div Σ e− λ

2 r2∇Σr 2

dσ = 2Re− λ

2 R2

∂DR

∇Σrdµ. (13) Hence 1 −

• DR

HΣ2dσ nλ Vol(DR) = R n Vol(DR)

• ∂DR

∇Σrdµ = e

λ 2 R2

2n Vol(DR)

• ∂DR

div Σ e− λ

2 r2∇Σr 2

dσ (14)

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A rigidity theorem for self-shrinkers of MCF.

Proof of our results. Lemma 9.

Proof of Lemma 9 Finally, the proposition follows taking into account in equation above that div Σ e− λ

2 r2∇Σr 2

=2e− λ

2 r2

n − λr 2 (15)

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A rigidity theorem for self-shrinkers of MCF.

The bound for the norm of second fundamental form in Theorem of Cao and Li is sharp

The bound λ is sharp in the following sense: Consider the non-compact and proper 1-self-shrinker given by Σ = Γp,q × R ⊆ R4, where Γp,q ⊆ R2 is an Abresch-Langer curve. We have that AR4

Σ 2 = AR2 Γ 2 = (kΓ g )2

where kΓ

g is the geodesic curvature (= signed curvature) of

the Abresch-Langer curve Γp,q ⊆ R2.

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A rigidity theorem for self-shrinkers of MCF.

The bound for the norm of second fundamental form in Theorem of Cao and Li is sharp

But the Abresch-Langer curves Γp,q are contained in an annulus around the origin, and they are curves with rotation number p which touches each boundary of the annulus q times for each pair of mutally prime positive integers p, q such that 1

2 < p q < √ 2 2

As it is shown in the following picture

• V. Palmer, UJI, Castell´
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A rigidity theorem for self-shrinkers of MCF.

The bound for the norm of second fundamental form in Theorem of Cao and Li is sharp

It has been shown, (see H. Halldorson, Trans. Amer. Math.

• Soc. 364, (2012)), that the signed curvature kΓ

g of Γp,q:

Is an increasing function of the radius, Never changes sign and Takes its maximum and minimum at the same time as the radius, kΓ

min = rmin and kΓ max = rmax, where rmin and rmax are

the inner and the outer radius of the annulus respectively. Moreover, rmin take on every value in (0, 1] and rmax take on every value in [1, ∞)

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.

The bound for the norm of second fundamental form in Theorem of Cao and Li is sharp

Hence, we can choose the values p and q in order to have:

AR4

Σ 2 = (kΓ g )2 < 5 3

The inner radius satisfies rmin < 1, so the sphere S3(1) separates Σ

In conclusion, if we consider bounds for ARn+m

Σ

2 greater than λ, there are λ-self-shrinkers satisfying this bound which are not those identified by Cao and Li in their Theorem. Moreover, some of these self-shrinkers can be separated.

• V. Palmer, UJI, Castell´
• joint work with: V. Gimeno, UJI

A rigidity theorem for self-shrinkers of MCF.